We prove that for any graph $G$ of maximum degree at most $\Delta$, the zeros of its chromatic polynomial $\chi_G(x)$ (in $\mathbb{C}$) lie inside the disc of radius $5.94 \Delta$ centered at $0$. This improves on the previously best known bound of approximately $6.91\Delta$. We also obtain improved bounds for graphs of high girth. We prove that for every $g$ there is a constant $K_g$ such that for any graph $G$ of maximum degree at most $\Delta$ and girth at least $g$, the zeros of its chromatic polynomial $\chi_G(x)$ lie inside the disc of radius $K_g \Delta$ centered at $0$, where $K_g$ is the solution to a certain optimization problem. In particular, $K_g < 5$ when $g \geq 5$ and $K_g < 4$ when $g \geq 25$ and $K_g$ tends to approximately $3.86$ as $g \to \infty$. Key to the proof is a classical theorem of Whitney which allows us to relate the chromatic polynomial of a graph $G$ to the generating function of so-called broken-circuit-free forests in $G$. We also establish a zero-free disc for the generating function of all forests in $G$ (aka the partition function of the arboreal gas) which may be of independent interest.

翻译：我们证明：对于任意最大度数至多为 $\Delta$ 的图 $G$，其色多项式 $\chi_G(x)$（在 $\mathbb{C}$ 中）的零点均位于以 $0$ 为圆心、半径 $5.94\Delta$ 的圆盘内。这一结果改进了此前已知的最佳界（约 $6.91\Delta$）。我们还得到了高围长图的改进界：对每个 $g$，存在常数 $K_g$，使得任意最大度数至多为 $\Delta$ 且围长至少为 $g$ 的图 $G$，其色多项式 $\chi_G(x)$ 的零点均位于以 $0$ 为圆心、半径 $K_g\Delta$ 的圆盘内，其中 $K_g$ 是某个优化问题的解。特别地，当 $g \geq 5$ 时 $K_g < 5$，当 $g \geq 25$ 时 $K_g < 4$，且当 $g \to \infty$ 时 $K_g$ 趋近于约 $3.86$。证明的关键在于 Whitney 的一个经典定理，该定理将图 $G$ 的色多项式与 $G$ 中所谓的无破圈森林的生成函数联系起来。我们还建立了 $G$ 中所有森林的生成函数（即树状气体配分函数）的一个无零点圆盘，这一结果可能具有独立的研究意义。